As \(t\rightarrow -\infty \) ,
\(\Omega _t\)  is approximable by the set \(\lbrace \log \left(c_{\beta }(z_0)|z-z_0|\right)<t \rbrace  \)  (see [1]}, [2]}). This implies
\(\lim _{t\rightarrow -\infty } \frac{e^{2t}}{ v{(\Omega _t)}} =\lim _{t\rightarrow -\infty } \frac{e^{2t}}{ v{(\lbrace |z-z_0|< {e^t c_{\beta }^{-1}(z_0) } \rbrace )}} =\frac{c_{\beta }^2(z_0)}{\pi }.\) 
